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RELATIVISTIC GLOBAL NAVIGATION SYSTEMANDREJA GOMBOC UROŠ KOSTIĆ MARTIN HORVAT SANTE CARLONI PACOME DELVA
PECS Mini-Workshop: Relativistic Positioning System, ESTEC, Sep 25th, 2014
GNSS - BASICS present GNSS:
• user receives “clock” information from 4 satellites
+ orbital parameters (determined by ground tracking) : :
positions of satellites user’s position
• 4 satellites (4D=3 space + 1 time)
• absolute space and time: reference frame (?) + relativistic corrections
alternative approach:
• in relativity: space and time are not absolute, proper times of satellites run at different paces, but each uniquely defines satellite’s position along its orbit
• instead of (x,y,z,t) use satellites’ proper times or emission coordinates (𝜏1, 𝜏2, 𝜏3, 𝜏4) + orbital parameters (determined by ground tracking) :
positions of satellites user’s position
EMISSION COORDINATES - ADVANTAGES
• physical quantities (proper times at the moment of the emission of the signal) measured on-board
• independent of any terrestrial reference frame
• relativistic effects already (naturally) included
• no need to synchronise satellite clocks
ABC CONCEPT
• on-board receivers (inter-satellite links)
• if the user is one of the satellites - from its own and proper times of 3 other satellites it can determine its own position at any given time
• NEXT STEP: let two satellites communicate their proper times to each other for some time
• pairs of (𝜏1, 𝜏2) allow deduction of satellites’ own orbital parameters!
• ground tracking not necessary
• Autonomous Basis of Coordinates (ABC)
ABC CONCEPT - ADVANTAGES
• robustness of recovering orbital parameters with respect to noise in the data
• consistency of description with redundant number of satellites
• its realisation does not depend on observations from Earth
• no entanglement with Earth internal dynamics, no Earth stations for maintaining of reference frame
• stability and accuracy
• based on well-known satellite dynamics, satellite orbits are very stable in time, and can be accurately described
• applications in science
• geophysics, relativistic gravitation and reference frames, determine/refine values of gravitational parameters (e.g. multipoles)?
ARIADNA PROJECTS
• gravitational field of isolated, spherically symmetric Earth
• Ariadna project by Čadež et al. 2010: relativistic positioning system is feasible, stable and accurate (transformation between x,y,z,t and emission coordinates works)
• Ariadna project by Čadež et al. 2011: inter-satellite communication and orbital parameters determination - ABC concept works
PECS RGNSS PROJECT 2011-2014
• more realistic case: non-spherically symmetric Earth + other celestial bodies
• all gravitational perturbations: Earth multipoles, tides, Moon, Sun, Jupiter Venus, Kerr effect (due to Earth rotation)
• do positioning and ABC concept still work?
Montenbruck and Gill, 2005
WORK PACKAGE 1 • built a mathematical scheme for including all relevant gravitational perturbations
to linear terms in general relativity
!
WORK PACKAGE 2 • satellite dynamics: calculated perturbed satellite orbits (slow time evolution of
orbital parameters)
!
!
spac
e
time
WORK PACKAGE 3
• positioning - works! (relative accuracy 10-32 - 10-30 in t, 10-28 - 10-26 in x,y,z) (!)
• ABC system: initially, orbital parameters known only with limited accuracy,
using inter-satellite links (pairs 𝜏1, 𝜏2) - possible to refine orbital parameters to relative accuracy 10-22 - works!
• possible degeneracies between parameters: none found!
WORK PACKAGE 4!
• is it possible to refine values of gravitational parameters?
• theoretically yes, because there is a well defined minimum in the action S (see Uroš’s presentation)
• due to high accuracy of calculations, they became slow…
• possible problems in practice:
(multi-dimensional minimization
methods, non-gravitational
perturbations,…)
• scientific applications
SUMMARY OF RESULTS
• relativistic positioning and ABC system in the realistic gravitational field (with gravitational perturbations) are (numerically) feasible, accurate and stable
• theoretically possible to “measure” the gravitational field of the Earth and nearby celestial bodies - independent way to measure space-time in the vicinity of Earth - various scientific applications
main points:
• accuracy, stability and lower costs of such a system - no ground tracking (link between a terrestrial reference frame and ABC established by several receivers at known terrestrial positions)
• no clock synchronisation
• independent, robust, consistent
• very promising!
POSSIBLE NEXT STEPS
• study of suitable methods for highly accurate and faster minimisation (Uroš)
• study of influence of non-gravitational perturbations on relativistic positioning and ABC
• tests on real satellite data (satellite dynamics, effects of non-gravitational perturbations?…)
• feasibility study on ground-based infrastructure and on-board hardware required for implementing ABC (put receivers on 2 Galileo satellites?)
• feasibility study of a system of small satellites with inter-satellite links - ABC concept demonstration, preceded by demonstration on the ground?
• …